Hans Havermann>Back in 2004, Peter Luschny wrote an open letter to Donald E. Knuth about his conviction that the proper definition of the Bernoulli number B(1) should be 1/2, not -1/2. Professor Knuth declined to embrace the idea. http://www.luschny.de/math/zeta/OpenLetter.pdf Peter has now put up a detailed response to Knuth, calling it The Bernoulli Manifesto. http://luschny.de/math/zeta/The-Bernoulli-Manifesto.html -------------- Wow, I expected DEK's only defense of the status quo would be "It's too late to change." But they've already changed at least once. Whittaker and Watson's nth Bernoulli number was |B_2n|. Date: Sun, 20 Feb 1994 02:54-0800 From: Bill Gosper <rwg@NEWTON.macsyma.com> Subject: Bernoulli numbers and polynomials To: conway@math.Princeton.EDU Cc: math-fun@cs.arizona.edu In-Reply-To: <19931017045155.6.RWG@TSUNAMI.macsyma.com> Message-Id: <19940220105422.2.RWG@SWEATHOUSE.macsyma.com> Date: Sat, 16 Oct 1993 21:51 PDT From: Bill Gosper <rwg@NEWTON> . . . JHC> We won't win on B^1 for a very long time. Which leads to my modest proposal: * Outlaw Bernoulli numbers altogether! * This is founded on my previously stated . . . religious conviction that any formula involving Bernoulli numbers can be elegantly generalized to instead use Bernoulli polynomials of a new variable. Then our only concession is always having to write B(x+1). (Like Gamma(x+1).) Several years ago I wrote these sentiments to Rich and Hilarie, and included some polynomialized formulas traditionally given with mere Bernoulli numbers. One I'd especially like to recover, ascribed to Rademacher, was posed as a challenge by N.J.A. Sloane in reponse to the polys-only proposal. Found it. 26 Nov 1980, (claimed) cc: D&E Lehmer, R Askey, N Sloane, L Guibas. But I think we already knew this: k k k k k k y y - (y + 1) y - 2 (y + 1) + (y + 2) B (y) = -- + ------------- + -------------------------- + . . . k 1 2 3 Anyway, it directly extends B (a) 1-s zeta(s, a) = ------- s - 1 to negative s. (And indirectly to conjectures like 3 zeta(3) - --------- = 2 4 pi 0 1 1 2 2 3 ----- log(1) + ----- (log(1) - log(2)) + ----- (log(1) - 2 log(2) + log(3)) 2 3 4 3 4 5 4 5 6 + . . ., later confirmed.) Other gossip therein: 7 zeta(3) --------- inf 2 /===\ 1/12 - n 8 pi | | e e | | ---------------------- = ---------- | | 1 2 (n - 1/3) n 5/36 n = 1 (1 - ---) 2 2 n and denom(B ) = 2 Fib(20). 40 --------------- rwg And I do NOT advocate τ over 2π.